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Lie theory for nilpotent L∞algebras
 Ann. Math
"... Let R be a commutative algebra over a field K of characteristic 0. The spectrum Spec(R) of R is the set Hom(R, K) of all homomorphisms from R to K. Let Ω • be the simplicial differential graded (dg) commutative algebra whose nsimplices Ωn are the dg algebra of differential forms on the geometric n ..."
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Let R be a commutative algebra over a field K of characteristic 0. The spectrum Spec(R) of R is the set Hom(R, K) of all homomorphisms from R to K. Let Ω • be the simplicial differential graded (dg) commutative algebra whose nsimplices Ωn are the dg algebra of differential forms on the geometric nsimplex ∆ n. In [20], Sullivan reformulated Quillen’s
HZalgebra spectra are differential graded algebras
 Amer. Jour. Math
, 2004
"... Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZalgebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZalgebra spectra. We also construct Qu ..."
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Cited by 63 (17 self)
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Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZalgebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZalgebra spectra. We also construct Quillen equivalences between the differential graded modules and module spectra over these algebras. We use these equivalences in turn to produce algebraic models for rational stable model categories. We show that bascially any rational stable model category is Quillen equivalent to modules over a differential graded Qalgebra (with many objects). 1.
DG coalgebras as formal stacks
 J. Pure Appl. Algebra
"... 1.1. In this paper we provide the category of unital (dg, unbounded) coalgebras dgcu(k) ..."
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Cited by 51 (4 self)
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1.1. In this paper we provide the category of unital (dg, unbounded) coalgebras dgcu(k)
Equivalences of monoidal model categories
 Algebr. Geom. Topol
, 2002
"... Abstract: We construct Quillen equivalences between the model categories of monoids (rings), modules and algebras over two Quillen equivalent model categories under certain conditions. This is a continuation of our earlier work where we established model categories of monoids, modules and algebras [ ..."
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Cited by 49 (10 self)
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Abstract: We construct Quillen equivalences between the model categories of monoids (rings), modules and algebras over two Quillen equivalent model categories under certain conditions. This is a continuation of our earlier work where we established model categories of monoids, modules and algebras [SS00]. As an application we extend the DoldKan equivalence to show that the model categories of simplicial rings, modules and algebras are Quillen equivalent to the associated model categories of connected differential graded rings, modules and algebras. We also show that our classification results from [SS] concerning stable model categories translate to any one of the known symmetric monoidal model categories of spectra. 1.
Descent of Deligne groupoids
 Int. Math. Res. Notices
, 1997
"... Abstract. To any nonnegatively graded dg Lie algebra g over a field k of characteristic zero we assign a functor Σg: art/k → Kan from the category of commutative local artinian kalgebras with the residue field k to the category of Kan simplicial sets. There is a natural homotopy equivalence betwee ..."
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Cited by 48 (3 self)
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Abstract. To any nonnegatively graded dg Lie algebra g over a field k of characteristic zero we assign a functor Σg: art/k → Kan from the category of commutative local artinian kalgebras with the residue field k to the category of Kan simplicial sets. There is a natural homotopy equivalence between Σg and the Deligne groupoid corresponding to g. The main result of the paper claims that the functor Σ commutes up to homotopy with the ”total space ” functors which assign a dg Lie algebra to a cosimplicial dg Lie algebra and a simplicial set to a cosimplicial simplicial set. This proves a conjecture of Schechtman [S1, S2, HS3] which implies that if a deformation problem is described “locally ” by a sheaf of dg Lie algebras g on a topological space X then the global deformation problem is described by the homotopy Lie algebra RΓ(X, g). 1.
Homotopical Algebraic Geometry I: Topos theory
, 2002
"... This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use Scategories (i.e. simplicially enriched categories) as models for certain kind of ..."
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Cited by 46 (18 self)
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This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos. For this, we use Scategories (i.e. simplicially enriched categories) as models for certain kind of ∞categories, and we develop the notions of Stopologies, Ssites and stacks over them. We prove in particular, that for an Scategory T endowed with an Stopology, there exists a model
LusternikSchnirelmann category and systolic category of low dimensional manifolds
 COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS
, 2004
"... We show that the geometry of a Riemannian manifold (M, G) is sensitive to the apparently purely homotopytheoretic invariant of M known as the LusternikSchnirelmann category, denoted catLS(M). Here we introduce a Riemannian analogue of catLS(M), called the systolic category of M. It is denoted cat ..."
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Cited by 27 (9 self)
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We show that the geometry of a Riemannian manifold (M, G) is sensitive to the apparently purely homotopytheoretic invariant of M known as the LusternikSchnirelmann category, denoted catLS(M). Here we introduce a Riemannian analogue of catLS(M), called the systolic category of M. It is denoted catsys(M), and defined in terms of the existence of systolic inequalities satisfied by every metric G, as initiated by C. Loewner and later developed by M. Gromov. We compare the two categories. In all our examples, the inequality catsys M ≤ catLS M is satisfied, which typically turns out to be an equality, e.g. in dimension 3. We show that a number of existing systolic inequalities can be reinterpreted as special cases of such equality, and that both categories are sensitive to Massey products. The comparison with the value of catLS(M) leads us to prove or conjecture new systolic inequalities on M.